The vector field \(\mathcal {M}_\Phi ^\Pi \)  is the modular vector field of \((M,\Pi )\)  with respect to \(\Phi \) . It is proved in [1]} that \(\mathcal {M}_\Phi ^\Pi \)  is a Poisson vector field (see (REF )).
In addition, if \(\Phi ^{\prime }=e^F \Phi \)  is another positive volume form on \(M\) , then the relation between \(\mathcal {M}_\Phi ^\Pi \)  and \(\mathcal {M}_{\Phi ^{\prime }} ^\Pi \)  is
\(\mathcal {M}_{\Phi ^{\prime }}^\Pi =\mathcal {M}_\Phi ^\Pi - X^\Pi _F.\) 
